o 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
main point of difference between the two cases can be seen, almost without 
mathematical analysis, as follows :— 
§ 2. Speaking somewhat loosely, the stability or instability may be measured by 
the resultant of several factors. In the case of an incompressible liquid we may sav 
that gravitation tends to stability, and rotation to instability; the liquid becomes 
unstable as soon as the second factor preponderates over the first. The gravitational 
tendency to stability arises in this case from the surface inequalities caused by the 
displacement: matter is moved from a place of higher potential to a place of lower 
potential, and in this way the gravitational potential energy is increased. As soon 
as we pass to the consideration of a compressible gas the case is entirely different. 
Suppose, to take the simplest case, that we are dealing with a single shell of 
gravitating gas, bounded by spheres of radii r and r -fi dr, and initially in equilibrium 
under its own gravitation, at a uniform density p 0 . 
Suppose, now, that this gas is caused to undergo a tangential compression or 
dilatation, such that the density is changed from 
Po fo Po “h — p«S, ; , 
where p„ is a small quantity, and S„ is a spherical surface harmonic of order n. 
It will readily be verified that there is a decrease in the gravitational energy of 
amount 
47rr 3 (drf t -- pn - - f f S/ sin 0 dd d<b. 
v ' (2 n + 1) J J 
As this is essentially a positive quantity, we see that any tangential displacement 
of a single shell will decrease the gravitational energy. 
This example is sufficient to show that when the gas is compressible, the tendency 
of gravitation may be towards instability. The gravitation of the surface inequalities 
will as before tend towards stability, but when we are dealing with a gaseous nebula, 
it is impossible to suppose that a discontinuity of density can occur such as would be 
necessary if this tendency were to come into operation. Rotation as before will tend 
to instability, and the factor which makes for stability will be the elasticity of 
the gas. 
We can now see that there is nothing inherently impossible, or even improbable, 
in the supposition that for a gaseous nebula the symmetrical configuration may 
become unstable even in the absence of rotation. The question which we shall 
primarily attempt to answer is, whether or not this is, in point of fact, a possible 
occurrence, and if so, under what circumstances it will take place. To investigate 
this problem, it will be sufficient to consider the vibrations of a non-rotating nebula 
about a configuration of spherical symmetry. 
§ 3. Unfortunately, the stability of a gaseous nebula of finite size is not a subject 
